Question

Find the second Taylor polynomial P2(x) for the function f (x) = ex cos x about x0 = 0.

a. Use P2(0.5) to approximate f (0.5). Find an upper bound for error |f (0.5) − P2(0.5)| using the error formula, and compare it to the actual error.
b. Find a bound for the error |f (x) − P2(x)| in using P2(x) to approximate f (x) on the interval [0, 1].
c. Approximate d. Find an upper bound for the error in (c) using and compare the bound to the actual error.

Answer 1

a.
`\( P_2(0.5) \)` approximates
`\( f(0.5) \)`, and the error
`\( |f(0.5) - P_2(0.5)| \)`
`\( |f(0.5) - P_2(0.5)| \)`can be upper-bounded using the error formula.

b.
`\( |f(x) - P_2(x)| \)`can be bounded on the interval [0, 1] when using
`\( P_2(x) \)`to approximate
`\( f(x) \).`

c.
`\( P_2(x) \)` approximates
`\( f(x) \)` on the interval
`[0, 1].`

d. An upper bound for the error in
`(c)`can be found and compared to the actual error.

Step-by-step explanation:

Taylor polynomials are used to approximate functions and provide insights into their behavior. In the context of the given function
`\( f(x) = e^x \cos x \) about \( x_0 = 0 \)`, the second Taylor polynomial
`\( P_2(x) \)`is employed for approximation.

a. Utilizing
`\( P_2(0.5) \)` as an approximation for
`\( f(0.5) \)`involves evaluating the polynomial at the given point. The error
`\( |f(0.5) - P_2(0.5)| \)` is then analyzed using the error formula, which provides an upper bound for the discrepancy between the actual and approximated values.

b. To estimate the error
`\( |f(x) - P_2(x)| \)`on the interval
`[0, 1]`, one considers the difference between the actual function and its second Taylor polynomial approximation.

c. The use of
`\( P_2(x) \)` to approximate
`\( f(x) \)` on the interval
`[0, 1]` is a broader application of the Taylor polynomial for gaining insights into the function's behavior within this range.

d. Determining an upper bound for the error in part (c) involves assessing the maximum discrepancy between the actual and approximated values within the specified interval. This bound is then compared to the actual error to evaluate the accuracy of the approximation.

Answer 2

To find the second Taylor polynomial for f(x) = ex cos x about x0 = 0, we find the first two derivatives of the function and evaluate them at x = 0. Using the Taylor series formula, the second Taylor polynomial P2(x) is 1 + x^2/2. To approximate f(0.5), we substitute x = 0.5 in P2(x), and the upper bound for the error is e/48.

Step-by-step explanation:

To find the second Taylor polynomial, we need to find the first two derivatives of the function and evaluate them at x = 0. For f(x) = ex cos x, the first derivative is f'(x) = ex (cos x - sin x) and the second derivative is f''(x) = ex (-2 sin x). Evaluating these at x = 0 gives f'(0) = 1 and f''(0) = 0.

Using the Taylor series formula, the second Taylor polynomial P2(x) is given by:

P2(x) = f(0) + f'(0)x + f''(0)x^2/2 = 1 + x^2/2.

a. To approximate f(0.5), we substitute x = 0.5 in P2(x):
P2(0.5) = 1 + (0.5)^2/2 = 1.125.

Using the error formula |f(0.5) - P2(0.5)| <= M/(3!)(0.5)^3, where M is an upper bound for the absolute value of the third derivative, we can find the upper bound for the error. In this case, the third derivative is f'''(x) = ex (-3 cos x), and since |ex (-3 cos x)| <= ex for all x, we can use e as our value for M. Evaluating the error formula gives an upper bound of e/(3!)(0.5)^3 = e/48.